# Statistical inverse learning problems with random observations

arXiv: https://arxiv.org/abs/2312.15341 · HTML: https://mlsanity.com/paper/2312.15341/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 82 references that are in the map.
- First posted on arXiv 23 Dec 2023.

## Builds on (most cited first, 10 of 10)

- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems](https://mlsanity.com/paper/1902.05404/) (2019)
- [Convex regularization in statistical inverse learning problems](https://mlsanity.com/paper/2102.09526/) (2021)
- [A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models](https://mlsanity.com/paper/1705.06079/) (2017)
- [Inverse learning in Hilbert scales](https://mlsanity.com/paper/2002.10208/) (2020)
- [Stochastic Gradient Descent in Hilbert Scales: Smoothness, Preconditioning and Earlier Stopping](https://mlsanity.com/paper/2006.10840/) (2020)
- [On Learning the Optimal Regularization Parameter in Inverse Problems](https://mlsanity.com/paper/2311.15845/) (2023)
- [Large Noise in Variational Regularization](https://mlsanity.com/paper/1602.00520/) (2016)
- [Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses](https://mlsanity.com/paper/2011.07539/) (2020)

## Built on by (most cited first, 1 of 1)

- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)

## Abstract (arXiv)

We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable solutions. We discuss recent results in spectral regularization methods and regularization by projection, exploring both approaches within the context of Hilbert scales and presenting new insights particularly in regularization by projection. Additionally, we overview recent advancements in regularization using convex penalties. Convergence rates are analyzed in terms of the sample size in a probabilistic sense, yielding minimax rates in both expectation and probability. To achieve these results, the structure of reproducing kernel Hilbert spaces is leveraged to establish minimax rates in the statistical learning setting. We detail the assumptions underpinning these key elements of our proofs. Finally, we demonstrate the application of these concepts to nonlinear inverse problems in pharmacokinetic/pharmacodynamic (PK/PD) models, where the task is to predict changes in drug concentrations in patients.